Of bodies chang'd to various forms, I sing:
Ye Gods, from whom these miracles did spring,
Inspire my numbers with celestial heat;
'Till I my long laborious work complete:
And add
perpetual tenour to my rhymes,
Deduc'd from Nature's birth, to Caesar's times.

"Dregs" refers to the lowest part--like the dregs of society. Here, "dregs" doesn't seem so low, because once purged of its ponderous ("having great mass and weight and unwieldiness") parts, it can ascend to Heaven as stars.

High o'er the clouds, and empty realms of wind,
The God a clearer space for Heav'n design'd;
Where fields of light, and liquid aether flow;
Purg'd from the pondrous
dregs of Earth below.

Thus, while the mute creation downward bend
Their sight, and to their earthly mother tend,
Man looks aloft; and with erected eyes
Beholds his own hereditary skies.
From such rude principles our form began;
And earth was metamorphos'd into Man.

In this section, Ovid describes the different ages (golden, silver, bronze, iron). "Brazen" refers to the bronze age, but the example sentence suggests that the word is also being used to describe the nature of the men during that age--their warlike rage could be "unrestrained by convention or propriety" but not yet to the point of being disrespectful to the gods.

To this came next in course, the
brazen age:
A warlike offspring, prompt to bloody rage,
Not impious yet...

"Promiscuous" also means "casual and unrestrained in sexual behavior"--this would be a fitting description of Phoebus/Apollo that sets the mood for the next story. But the example sentence is using the word to describe how every plant that is green has been used to decorate the hair of both Apollo and the winners of the Pythian games.

But every green alike by Phoebus worn,
Did, with
promiscuous grace, his flowing locks adorn.

"Espouse" also means "take in marriage" but since Daphne is now a tree, Apollo can't marry her, so instead, he chooses her to be the symbol of victory, which seems odd, since he was defeated in her pursuit of her and she didn't exactly get the help she wanted. But "the grateful tree was pleas'd with what he said; And shook the shady honours of her head."

Because thou canst not be
My mistress, I
espouse thee for my tree:
Be thou the prize of honour, and renown;
The deathless poet, and the poem, crown.

Thus Argus lies in pieces, cold, and pale;
And all his hundred eyes, with all their light,
Are clos'd at once, in one perpetual night.
These Juno takes, that they no more may fail,
And spreads them in her peacock's
gaudy tail.